The Five Brigands

The five Spanish brigands, Alfonso, Benito, Carlos, Diego, and Esteban, were counting their spoils after a raid, when it was found that they had captured altogether exactly 200 doubloons. One of the band pointed out that if Alfonso had twelve times as much, Benito three times as much, Carlos the same amount, Diego half as much, and Esteban one-third as much, they would still have altogether just 200 doubloons. How many doubloons had each? There are a good many equally correct answers to this question. Here is one of them: A 6 × 12 = 72 B 12 × 3 = 36 C 17 × 1 = 17 D 120 × ½ = 60 E 45 × 1/3 = 15 ___ ___ 200 200 The puzzle is to discover exactly how many different answers there are, it being understood that every man had something and that there is to be no fractional money--only doubloons in every case. This problem, worded somewhat differently, was propounded by Tartaglia (died 1559), and he flattered himself that he had found one solution; but a French mathematician of note (M.A. Labosne), in a recent work, says that his readers will be astonished when he assures them that there are 6,639 different correct answers to the question. Is this so? How many answers are there?